Construction of Permutation Tests
Journal of the American Statistical Association, 85 (411):
693--698 (1990)Abstract
A general method for constructing permutation tests for
various experimental designs follows from invariance and
sufficiency. In this framework, randomization (or
rerandomization) tests are just special cases of
permutation tests. The methodology extends the
applicability of permutation tests: An example demonstrates
a test for an interaction effect in a factorial experiment,
a problem thought to be unapproachable by permutation or
randomization methods. Tests are constructed as follows.
The modeling of block and treatment effects closely
parallels classical analysis of variance (ANOVA). The ANOVA
assumptions of iid normal random errors are replaced,
however, by the much weaker assumption of exchangeability
of subsets of random errors. Distributional dependences on
the unknown nuisance parameters for blocks, covariates, and
untested treatments and on the unknown error distribution
are eliminated by invariance and sufficiency, respectively.
Initial reduction of the data by invariance appears to
extend considerably the applicability of permutation tests.
Because of the assumed exchangeability of the errors, the
sufficiency step leads to data permutations when computing
the test. The method also addresses some philosophical
problems associated with randomization tests. For instance,
in this article's framework, it is the design, and not the
data, which is always regarded as fixed, so ancillarity of
the design causes no difficulty.